Rastmic rank-3 clan

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The rastmic rank-3 clan of temperaments tempers out the rastma, 243/242. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an 11/9 neutral third.

Temperaments discussed elsewhere include:

Jove (+441/440 or 540/539) → Breed family
Hagrid (+9801/9800) → Cataharry family

Considered below are spectacle, mirwomo, mandos, cuckoo, parahemif, urania, rabic, and mirage.

Spectacle

Spectacle, named by Gene Ward Smith in 2010^[1], can be described as the 31 & 34d & 41 temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.

It is associated with the marvo temperamment.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 4 5], ⟨0 0 1 2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

WE: ~2 = 1200.5486 ¢, ~11/9 = 350.2171 ¢, ~5/4 = 384.1078 ¢

error map: ⟨+0.549 -0.972 -1.109 +0.806 +0.864]

CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.1758 ¢, ~5/4 = 384.0951 ¢

error map: ⟨0.000 -1.603 -2.219 +0.068 -0.439]

Minimax tuning:

11-odd-limit: ~2 = [1 0 0 0 0⟩, ~11/9 = [-2/5 0 0 0 1/5⟩, ~5 = [2/5 -2 1 0 4/5⟩

unchanged-interval (eigenmonzo) basis: 2.9/5.11

Optimal ET sequence: 24d, 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc

Badness (Sintel): 0.599

Projection pairs: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9

Scales: spectacle31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 4 5 -2], ⟨0 0 1 2 0 4]]

Optimal tunings:

WE: ~2 = 1200.6024 ¢, ~11/9 = 350.1004 ¢, ~5/4 = 384.5435 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.0393 ¢, ~5/4 = 384.5866 ¢

Optimal ET sequence: 31, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff *

* optimal patent val: 240

Badness (Sintel): 0.944

Mirwomo

For the 7-limit version, see Miscellaneous 7-limit temperaments #Mirwomo.

Mirwomo tempers out 385/384 and may be described as the 24 & 31 & 41 temperament, equating the undecimal quartertone ~33/32 with the septimal quartertone ~36/35.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 -3 5], ⟨0 0 1 -1 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

WE: ~2 = 1200.7360 ¢, ~11/9 = 350.1700 ¢, ~5/4 = 384.3403 ¢

error map: ⟨+0.736 -0.879 -0.501 -0.733 +1.004]

CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.0035 ¢, ~5/4 = 384.0785 ¢

error map: ⟨0.000 -1.948 -2.235 -2.915 -1.301]

Optimal ET sequence: 17, 21e, 24, 31, 41, 72, 247c, 312bd, 384bcdd, 456bcdde, 528bcdde, 631bbccdde

Badness (Sintel): 0.770

Mandos

Mandos tempers out 176/175 and may be described as the 24 & 27e & 31 temperament.

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

WE: ~2 = 1199.1949 ¢, ~11/9 = 350.6135 ¢, ~5/4 = 390.4090 ¢

error map: ⟨-0.805 -1.533 +2.485 +1.814 +0.139]

CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5548 ¢, ~5/4 = 390.2690 ¢

error map: ⟨0.000 -0.845 +3.955 +3.410 +1.456]

Optimal ET sequence: 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde

Badness (Sintel): 0.902

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 0 5 5 -1], ⟨0 0 1 -2 0 0]]

Optimal tunings:

WE: ~2 = 1198.5555 ¢, ~11/9 = 351.0300 ¢, ~5/4 = 391.0458 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.1554 ¢, ~5/4 = 391.1227 ¢

Optimal ET sequence: 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff

Badness (Sintel): 0.863

Cuckoo

Cuckoo, named by Johannes Werpup in 2014^[2], tempers out 126/125 and may be described as the 24d & 27e & 31 temperament.

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 -4 5], ⟨0 0 1 3 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

WE: ~2 = 1199.8222 ¢, ~11/9 = 350.4356 ¢, ~5/4 = 389.8478 ¢

error map: ⟨-0.178 -1.262 +3.178 -1.558 +0.504]

CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4213 ¢, ~5/4 = 389.7308 ¢

error map: ⟨0.000 -1.112 +3.417 -1.318 +0.788]

Optimal ET sequence: 24d, 27e, 31, 58, 89, 154, 185

Badness (Sintel): 1.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]

Optimal tunings:

WE: ~2 = 1199.7103 ¢, ~11/9 = 350.5840 ¢, ~5/4 = 389.8071 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5682 ¢, ~5/4 = 389.6104 ¢

Optimal ET sequence: 27e, 31, 58, 96d, 154

Badness (Sintel): 1.23

Parahemif

For the 7-limit version, see Miscellaneous 7-limit temperaments #Parahemif.

Parahemif tempers out 896/891 and may be described as the 24 & 34d & 41 temperament. It is related to hemif, the no-5 rank-2 temperament that tempers out the same list of commas. As such, it finds the interval class of 7 at +13 generator steps, as a semi-augmented sixth (C–At). In the 13-limit, it tempers out 144/143, 352/351, 364/363 among others, and finds ~16/13 at the same neutral third as ~11/9.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891

Mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5], ⟨0 0 1 0 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

WE: ~2 = 1199.2633 ¢, ~11/9 = 351.3189 ¢, ~5/4 = 387.7835 ¢

error map: ⟨-0.737 -0.054 -0.004 -0.944 +3.803]

CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.4593 ¢, ~5/4 = 387.4226 ¢

error map: ⟨0.000 +0.964 +1.109 +0.145 +5.979]

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99e *

* optimal patent val: 123

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 243/242, 364/363

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1], ⟨0 0 1 0 0 0]]

Optimal tunings:

WE: ~2 = 1198.7603 ¢, ~11/9 = 351.3275 ¢, ~5/4 = 388.7872 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.6042 ¢, ~5/4 = 388.4720 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff

Badness (Sintel): 1.12

Urania

Urania tempers out 81/80, the syntonic comma. It is essentially mohaha with an independent generator for prime 7.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120

Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]

mapping generators: ~2, ~11/9, ~7

Mapping to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]

Lattice basis:

11/9 length = 0.2536, 8/7 length = 2.807

Angle (11/9, 8/7) = 90 degrees

Optimal tunings:

WE: ~2 = 1201.8548 ¢, ~11/9 = 348.6318 ¢, ~5/4 = 965.0936 ¢

error map: ⟨+1.855 -2.836 +2.741 -0.023 -4.449]

CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.3793 ¢, ~7/4 = 965.6304 ¢

error map: ⟨0.000 -5.196 +0.721 -3.196 -9.421]

Optimal ET sequence: 7d, 14c, 17c, 24, 31, 100de, 131bdee, 162bdee

Badness (Sintel): 1.01

Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5

Scales: urania24

Rabic

If the rastma is added to the list of commas along with the Alpharabian comma, you end up with rabic, which splits the octave into 24 equal parts. This temperament is named as such by Aura in 2022 because tempering out both the Alpharabian comma and the rastma automatically tempers out the Betarabian comma.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 131769/131072

Mapping: [⟨24 38 0 0 83], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

mapping generators: ~33/32, ~5, ~7

Optimal tunings:

WE: ~33/32 = 50.0220 ¢, ~5/4 = 385.0647 ¢, ~7/4 = 967.3158 ¢

error map: ⟨+0.538 -1.104 -0.002 -0.002 +0.541]

CWE: ~33/32 = 50.0000 ¢, ~5/4 = 385.0647 ¢, ~7/4 = 967.3158 ¢

error map: ⟨0.000 -1.955 -0.937 -1.133 -1.318]

Optimal ET sequence: 24, 48(d), 72, 264, 336b, 408b, 480bcde

Badness (Sintel): 7.30

Mirage

Mirage is miracle with an independent generator for prime 13.

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

mapping generators: ~2, ~15/14, ~13

Optimal tunings:

WE: ~2 = 1200.7626 ¢, ~15/14 = 116.7069 ¢, ~13/8 = 838.2364 ¢

error map: ⟨+0.763 -0.951 -0.974 +0.048 +0.810 -0.004]

CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6469 ¢, ~13/8 = 838.2123 ¢

error map: ⟨0.000 -2.074 -2.842 -2.120 -1.615 -2.315]

Optimal ET sequence: 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef

Badness (Sintel): 0.691

17-limit

Mirage is very naturally a 17-limit temperament, relating 13 and 17 by tempering out 273/272, 715/714, 833/832, and 936/935. Instead of 13/8, the second generator can also be the small comma tempered out by miraculous.

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384

Mapping: [⟨1 1 3 3 2 0 0], ⟨0 6 -7 -2 15 0 4], ⟨0 0 0 0 0 1 1]]

Optimal tunings:

WE: ~2 = 1200.7628 ¢, ~15/14 = 116.6995 ¢, ~13/8 = 837.1672 ¢
CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6395 ¢, ~13/8 = 837.1424 ¢

Optimal ET sequence: 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff

Badness (Sintel): 0.715